Non-uniform elliptic equations in convex Lipschitz domains

Non-uniform elliptic equations in convex Lipschitz domains are concerned. The non-smooth domains consist of a periodic connected high permeability sub-region and a periodic disconnected matrix block subset with low permeability. Let ϵ∈(0,1]ϵ∈(0,1] denote the size ratio of the matrix blocks to the whole domain and let ω2∈(0,1]ω2∈(0,1] denote the permeability ratio of the disconnected matrix block subset to the connected sub-region. The W1,pW1,p norm for p∈(1,∞)p∈(1,∞) of the elliptic solutions in the high permeability sub-region is shown to be bounded uniformly in ω,ϵω,ϵ. However, the W1,pW1,p norm of the solutions in the low permeability subset may not be bounded uniformly in ω,ϵω,ϵ. Roughly speaking, if the sources in the low permeability subset are small enough, the solutions in that subset are bounded uniformly in ω,ϵω,ϵ. Otherwise the solutions cannot be bounded uniformly in ω,ϵω,ϵ. Relations between the sources and the variation of the solutions in the low permeability subset are also presented in this work.